Chapter 1 Introduction
5.2 Modal Isolation
5.2.1 Rotation of Shaking and Recording Directions
For a perfectly symmetric dam with the shaker at the centerline, shaking in the stream direction excites only symmetric modes and shaking in the cross-stream di- rection excites only antisymmetric modes. This is because the motion of the dam centerline is in the stream direction for a symmetric mode and cross-stream for an antisymmetric mode. At Pacoima Dam, due to the lack of sufficient symmetry, the directions of motion at location C for the first symmetric mode and the first anti- symmetric mode were both primarily in the stream direction. As a result, there is
2.5 3 4 5 6 7 8 9 10 11 0
1 2 3 4 5 6x 10−3
Amplitude (Ranger Output / ω3 )
Frequency (Hz)
Figure 5.3: Frequency response curve for channel 1fv from the N85E shaking test considerable interference between the two modes for both directions of shaking, and this makes the determination of natural frequencies and damping difficult. Figure 5.3 shows the interfering resonances of the symmetric and antisymmetric modes between 5 Hz and 6 Hz in the amplitude of the channel 1fv response from the N85E shake.
The quantity plotted in the figure is the amplitude of the Ranger output divided by frequency cubed and is proportional to the displacement of the dam per unit shaker force. All frequency response curves in this chapter and in Appendix A have been plotted similarly.
One technique to eliminate interference between two modes is to align the direction of shaking perpendicular to the motion of one of the modes, which should eliminate the response of that mode, thus isolating the other one (Duron and Hall, 1986). For the Pacoima Dam data, this was done mathematically by combining the results of the two shaking directions vectorially. The symmetric mode was eliminated in this way. However, the procedure was less successful in eliminating the antisymmetric
mode, so it was modified by including a perpendicular force component with its phase shifted by 90◦. This method should be capable of eliminating a mode with motion that is elliptical at a point, which could be an effect of non-classical modes.
As a further enhancement, the pair of Ranger data channels at locations C, R and L were also combined vectorially in order to maximize the peak of the mode being isolated. The resultant Ranger output with amplitude A and phase θ measured in a direction rotated clockwise from radial by an angle α with shaking in a direction rotated clockwise from N85E (stream) by an angleβ and an elliptical shape with the minor axis a factor of C as large as the major axis is given by
Asin(ωt+θ) = [RSsin(ωt+φRS) cosα+TSsin(ωt+φT S) sinα] cosβ + [RXsin(ωt+φRX) cosα+TXsin(ωt+φT X) sinα] sinβ
+C[RSsin(ωt+φRS +γ) cosα+TSsin(ωt+φT S+γ) sinα] cos(β+ π 2) +C[RXsin(ωt+φRX +γ) cosα+TXsin(ωt+φT X+γ) sinα] sin(β+ π 2)
(5.1)
whereω is the excitation frequency,RS andTS are the measured radial and tangential amplitudes for the N85E (stream) shaking, RX and TX are the measured radial and tangential amplitudes for the S05E (cross-stream) shaking, φRS and φT S are the measured radial and tangential phases for the stream shaking, φRX and φT X are the measured radial and tangential phases for the cross-stream shaking, C is between 0 and 1 (C = 0 gives a unidirectional force) and γ gives the 90◦ phase shift of the perpendicular force component in order to get an elliptical force (γ = π2 or γ = −π2 for counterclockwise or clockwise rotation, respectively).
It was found that unidirectional shaking along S01E eliminated the symmetric mode in the vicinity of the resonating antisymmetric mode. To eliminate the anti- symmetric mode, best results were achieved with a shaking force at S21E combined with a 90◦ phase shifted and perpendicular force component 15% as large with coun- terclockwise rotation, although even this modified procedure does not appear to com- pletely eliminate the antisymmetric mode from the vicinity of the symmetric mode resonance.
4 4.5 5 5.5 6 6.5 7 0
0.5 1 1.5x 10−3
Frequency (Hz) Amplitude (Ranger Output / ω3 )
C R L
Figure 5.4: Frequency response curves on the crest at locations C, R and L for the antisymmetric mode
Results of the modal isolation attempt are shown in Figure 5.4 for the antisym- metric mode and Figure 5.5 for the symmetric mode. Motions at locations C, R and L are included in each figure. Because the shaker force has been rotated significantly from N85E, the response amplitudes are reduced 70% or more from those shown in Figure 5.3 for the N85E shake. From Figure 5.4, the resonant frequency of the an- tisymmetric mode is found to be around 5.65 Hz to 5.70 Hz with damping between 4.5% and 5.5% of critical, which was determined by the half-power method. As shown in Figure 5.5, the resonant frequency of the symmetric mode is between 5.30 Hz and 5.45 Hz depending on the location. The variation in natural frequency is probably a result of the antisymmetric mode not being completely eliminated. The damping for the symmetric mode estimated by the half-power method appears to be affected by the remaining presence of the antisymmetric mode, particularly at location C. The damping estimated from locations R and L is around 5.5% to 7.0% of critical.
The determined directions of motion and the phase shifts relative to location C at locations C, R and L for the first symmetric and first antisymmetric modes are
4 4.5 5 5.5 6 6.5 7 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 10−3
Frequency (Hz) Amplitude (Ranger Output / ω3 )
C R L
Figure 5.5: Frequency response curves on the crest at locations C, R and L for the symmetric mode
shown in Table 5.1. A negative relative phase indicates that the location lags behind the reference. It should be noted that the phase shifts for locations R and L are not particularly close to the values of 0◦ or 180◦ expected for classical modes. The measured mode shapes are plotted in Figure 5.6 using the amplitudes determined from Figures 5.4 and 5.5 and the directions given in Table 5.1. While there are some noticeable differences, the shapes have a basically similar character to those estimated by MODE-ID (Figure 4.1). At location C, the S87E direction of motion for the symmetric mode is 4◦ away from being perpendicular to the S01E shaking force orientation used to eliminate the symmetric mode; and the N63E direction of motion for the antisymmetric mode is 6◦ away from being perpendicular to the S21E shaking force orientation used to eliminate the antisymmetric mode. These differences could represent small errors in alignment of the shaker and Rangers, or be due to some modal interference still present, or they could result from some variation in motion of the dam that occurs over the 2.6 meter distance separating the shaker and location C.
Symmetric Mode Antisymmetric Mode Location Direction Phase Direction Phase
C S87E 0.0◦ N63E 0.0◦
R S29E -21.08◦ N64E -23.47◦
L S87E -23.85◦ S59E 144.35◦
Table 5.1: Direction of motion and relative phase for locations on the crest for the symmetric and antisymmetric modes
Symmetric
Antisymmetric
Figure 5.6: Symmetric and antisymmetric mode shapes determined from forced vi- bration testing (The open circles are the locations of the crest level Rangers.)